Data science, statistics or machine learning in broken English

In 2 previous posts, you learned what Bayesian modeling and Stan are and how to install them. Now you are ready to try it on some very Bayesian problems - as many people love - such as hierarchical Bayesian model.

Definition of hierarchical Bayesian models

Prior to tackling with a practical example, let's overview what and how hierarchical Bayesian model is. A famous book on Bayesian modeling with MCMC, written by Toshiro Tango and Taeko Becque and published in Japan, describes as below*1.

In a fixed-effects model of frequentist, each result is assumed to have a common average .

On the other hand, in a random-effects model, each result is assumed to have a distinct average and it is distributed around a global average .

Bayesian hierarchical models assume prior probability for parameters of a probability distribution of in a random-effects model, such as

It is said that such models have a hierarchical structure with two levels, that is,

1st level: a probability distribution is assumed for

2nd level: one more probability distribution is assumed for parameters of the 1st level

This is a textbook definition of hierarchical models, but I think it can be understood more intuitively; in hierarchical Bayesian models, often the models have to handle some excessive fluctuations as nonlinear effects more than expected in usual frequentist's models. Priors used in such models can be seen as an "absorber" that can absorb various kinds of fluctuations distributed around true parameters.

*1:Its original text is of course in Japanese, so this is just my own interpretation

Are you ready now? OK, this post reviews how to install Stan. Let's start here! :) In principle this post just follows a content of "RStan Getting Started" but some tips are added in order to fix less known problems.

Although I've written a series of posts titled "Machine Learning for package uses in R", usually I don't run machine learning on daily analytic works because my current coverage is so-called an ad-hoc analysis.

Instead of machine learning, ad-hoc analysts often use statistical modeling such as linear models (called "multiple regression" in general), generalized linear models (GLM) and/or econometric time series analysis. But in some situations such linear model and its variants would not work because of nonlinear components and/or individual variance, called "random effect".

In general, random effect can be well handled by generalized linear mixed models (GLMM) and for example CRAN has some related packages. But in some cases random effects cannot be formulated concisely and explicitly... if so, we have a strong alternative method to resolve it: "Bayesian using Markov Chain Monte Carlo (MCMC) method".

As one of the strongest methods for ad-hoc analysis, a series of posts will argue about Bayesian modeling with MCMC and its apllication. For the first time, this post overviews it.